In the present work, we solve the stabilization problem of a system of \(m\)  parabolic PDEs in cascade, where only the first equation is controlled and diffusion coefficients are distinct. We follow a modal decomposition approach. For internal control, we generalize methods mainly used for the scalar case (see [1]} on direct Lyapunov method for state feedback, see also [2]}) to the case of underactuated system with one scalar control. We assume that the number of internal inputs appearing in the first equation is equal to the number of unstable modes and that the resulting matrix that multiplies control inputs in the unstable modes is non-singular. We then introduce a novel state transformation for ODEs with dimension equal to the number of coupled PDEs and written as a polynomial matrix in the slower eigenvalues of a related Stürm-Liouville problem, with order related to the number of distinct diffusion coefficients. The coefficients of this polynomial matrix are nilpotent matrices up to identity matrix, which are subject to recursive generalized Sylvester equations and can be easily determined via a provided algorithm, while their values depend on the dynamics of the parabolic system. The stabilizing law simply consists in determining control gains stabilizing the reaction matrix and also in calculating our introduced state transformation, which depends on system dynamics. In this way, for any given system specification we provide a construction of unified and scalable control laws independently of the number of eigenvalues needed to be stabilized, which can be arbitrarily large.
Then for boundary control, we follow an indirect approach in order to place the controls internally and obtain a dynamic control law. This is relevant since the Hautus test might fail in the presence of distinct diffusion coefficients when trying to stabilize directly by use of static feedback. We assume that the control placed on the right boundary of the first state is written as a sum of control components. Inspired by the recent dynamic extension approach in [3]} for the scalar case, we adapt similar transformation to our vector case. The system is first mapped into a system where the control components and their time-derivatives are placed internally in the PDEs. We then apply modal decomposition followed by another transformation to the eigenspectrum in order to place control components in the first equation. In the next step, we are in a position to choose the dynamic control law. It turns out that for distributed control, we can achieve an arbitrarily fast decay rate while for boundary control, this is not the case. The present work solves the problem of internal and boundary stabilization of underactuated systems, for which backstepping approaches have not been proven to give solutions yet and at the same time it provides a scalable stabilization algorithm despite the presence of distinct diffusion coefficients.
